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Register a new userParallel Bayesian Optimization of Multiple Noisy Objectives with Expected Hypervolume Improvement
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Samuel Daulton
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Optimizing multiple competing black-box objectives is a challenging problem in many fields, including science, engineering, and machine learning. Multi-objective Bayesian optimization is a powerful approach for identifying the optimal trade-offs between the objectives with very few function evaluations. However, existing methods tend to perform poorly when observations are corrupted by noise, as they do not take into account uncertainty in the true Pareto frontier over the previously evaluated designs. We propose a novel acquisition function, NEHVI, that overcomes this important practical limitation by applying a Bayesian treatment to the popular expected hypervolume improvement criterion to integrate over this uncertainty in the Pareto frontier. We further argue that, even in the noiseless setting, the problem of generating multiple candidates in parallel reduces that of handling uncertainty in the Pareto frontier. Through this lens, we derive a natural parallel variant of NEHVI that can efficiently generate large batches of candidates. We provide a theoretical convergence guarantee for optimizing a Monte Carlo estimator of NEHVI using exact sample-path gradients. Empirically, we show that NEHVI achieves state-of-the-art performance in noisy and large-batch environments.
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In many real-world scenarios, decision makers seek to efficiently optimize multiple competing objectives in a sample-efficient fashion. Multi-objective Bayesian optimization (BO) is a common approach, but many of the best-performing acquisition functions do not have known analytic gradients and suffer from high computational overhead. We leverage recent advances in programming models and hardware acceleration for multi-objective BO using Expected Hypervolume Improvement (EHVI)---an algorithm notorious for its high computational complexity. We derive a novel formulation of q-Expected Hypervolume Improvement (qEHVI), an acquisition function that extends EHVI to the parallel, constrained evaluation setting. qEHVI is an exact computation of the joint EHVI of q new candidate points (up to Monte-Carlo (MC) integration error). Whereas previous EHVI formulations rely on gradient-free acquisition optimization or approximated gradients, we compute exact gradients of the MC estimator via auto-differentiation, thereby enabling efficient and effective optimization using first-order and quasi-second-order methods. Our empirical evaluation demonstrates that qEHVI is computationally tractable in many practical scenarios and outperforms state-of-the-art multi-objective BO algorithms at a fraction of their wall time.
In the field of multi-objective optimization algorithms, multi-objective Bayesian Global Optimization (MOBGO) is an important branch, in addition to evolutionary multi-objective optimization algorithms (EMOAs). MOBGO utilizes Gaussian Process models learned from previous objective function evaluations to decide the next evaluation site by maximizing or minimizing an infill criterion. A common criterion in MOBGO is the Expected Hypervolume Improvement (EHVI), which shows a good performance on a wide range of problems, with respect to exploration and exploitation. However, so far it has been a challenge to calculate exact EHVI values efficiently. In this paper, an efficient algorithm for the computation of the exact EHVI for a generic case is proposed. This efficient algorithm is based on partitioning the integration volume into a set of axis-parallel slices. Theoretically, the upper bound time complexities are improved from previously $O (n^2)$ and $O(n^3)$, for two- and three-objective problems respectively, to $Theta(nlog n)$, which is asymptotically optimal. This article generalizes the scheme in higher dimensional case by utilizing a new hyperbox decomposition technique, which was proposed by D{a}chert et al, EJOR, 2017. It also utilizes a generalization of the multilayered integration scheme that scales linearly in the number of hyperboxes of the decomposition. The speed comparison shows that the proposed algorithm in this paper significantly reduces computation time. Finally, this decomposition technique is applied in the calculation of the Probability of Improvement (PoI).
The Expected Improvement (EI) method, proposed by Jones et al. (1998), is a widely-used Bayesian optimization method, which makes use of a fitted Gaussian process model for efficient black-box optimization. However, one key drawback of EI is that it is overly greedy in exploiting the fitted Gaussian process model for optimization, which results in suboptimal solutions even with large sample sizes. To address this, we propose a new hierarchical EI (HEI) framework, which makes use of a hierarchical Gaussian process model. HEI preserves a closed-form acquisition function, and corrects the over-greediness of EI by encouraging exploration of the optimization space. We then introduce hyperparameter estimation methods which allow HEI to mimic a fully Bayesian optimization procedure, while avoiding expensive Markov-chain Monte Carlo sampling steps. We prove the global convergence of HEI over a broad function space, and establish near-minimax convergence rates under certain prior specifications. Numerical experiments show the improvement of HEI over existing Bayesian optimization methods, for synthetic functions and a semiconductor manufacturing optimization problem.
Bayesian Optimization is a sample-efficient black-box optimization procedure that is typically applied to problems with a small number of independent objectives. However, in practice we often wish to optimize objectives defined over many correlated outcomes (or ``tasks). For example, scientists may want to optimize the coverage of a cell tower network across a dense grid of locations. Similarly, engineers may seek to balance the performance of a robot across dozens of different environments via constrained or robust optimization. However, the Gaussian Process (GP) models typically used as probabilistic surrogates for multi-task Bayesian Optimization scale poorly with the number of outcomes, greatly limiting applicability. We devise an efficient technique for exact multi-task GP sampling that combines exploiting Kronecker structure in the covariance matrices with Matherons identity, allowing us to perform Bayesian Optimization using exact multi-task GP models with tens of thousands of correlated outputs. In doing so, we achieve substantial improvements in sample efficiency compared to existing approaches that only model aggregate functions of the outcomes. We demonstrate how this unlocks a new class of applications for Bayesian Optimization across a range of tasks in science and engineering, including optimizing interference patterns of an optical interferometer with more than 65,000 outputs.
Scarce data is a major challenge to scaling robot learning to truly complex tasks, as we need to generalize locally learned policies over different task contexts. Contextual policy search offers data-efficient learning and generalization by explicitly conditioning the policy on a parametric context space. In this paper, we further structure the contextual policy representation. We propose to factor contexts into two components: target contexts that describe the task objectives, e.g. target position for throwing a ball; and environment contexts that characterize the environment, e.g. initial position or mass of the ball. Our key observation is that experience can be directly generalized over target contexts. We show that this can be easily exploited in contextual policy search algorithms. In particular, we apply factorization to a Bayesian optimization approach to contextual policy search both in sampling-based and active learning settings. Our simulation results show faster learning and better generalization in various robotic domains. See our supplementary video: https://youtu.be/MNTbBAOufDY.
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